Welcome to Math 111Algebra in Business and Economics

Instructor: Alexandra Nichifor

Class Website:http://www.math.washington.edu/~nichifor/111F06.htm

Please pick up a copy of the syllabus and today’s handout

Worksheet 1: Speed as a Rate of Change

Time t (minutes)

Distance D travelled (miles)after time t

0

10

20

30

40

50

60

70

75

0120170

180

180

205

280

430

550

Q?: How far did the rocket travel from 50min to 60min?

A: 75 miles

205280280-205=75

Delta Notation:

• Greek letter Delta: is shorthand for“the change in”

Example:

How far did the rocket travel from 50min to 60min?

milesD 75205280

min)105060( tofdurationaover

Q: What was the average speed of the rocket from

t=50 min to t=60 min?

average speed = Change in distanceChange in time t

D

Shorthand(=change

in…)

5.71075

5060205280

tDAS miles per minute

On the graph:

75D

10t

Rise

Run

Moral: Average Speed (AS) from time t1 to time t2

=slope of the secant line thru the graph of the distance at points t1 & t2

A rate of change is a measure of how fast a quantity is changing with respect to time

Examples?

VIP Example: average speed is a rate of change of distance.

average speed = Change in distanceChange in time t

D

In general:

Average Rate of Change of Blah =tBlah

Change in Blah

Change in time

Types of Rates of Change:

Average Rate of Change Actual (Instantaneous) (will study in Math 112)

Example: Actual speed, as read off a speedometer

Overall (from t=0 to later time)

Incremental(from t=a to t=b)

Example: Average Trip Speed (ATS)

Example: Average Speed (AS)

tDAS

in time changedistancein change

tDATS

far so timefar so distance

• Note: An overall rate of change (such as ATS) is a special case of an incremental rate of change.

that is, one in which the initial time t1=0.

• Question: How do we measure the ATS on a graph of distance?For example, on our handout, what was the ATS over the first hour?

• Answer: Compute the slope of the line from the beginning of the graph (t1=0) to t2=60 min.

ATS=Slope of Diagonal

Line

Tip: Can pick any two points on this line to compute the slope!

Rise=350

Run=75Answer: ATS over the first hour = 350 / 75 = 4.67 mpm = 280 mphNote: Using the original two points: 280 / 60 =4.67 mpm

…To be continued on Friday…

Homework for Friday:

1. Familiarize yourself with the class rules by carefully reading the syllabus and the class website.

2. Print the Lecture Handouts (and bring to class!)

3. Get the text, a ruler, and a scientific calculator.

4. Read and do the problems in the Prologue.

5. Start working on Worksheet 1.